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Bounded Treewidth and Space-Efficient Linear Algebra [article]

Nikhil Balaji, Samir Datta
2014 arXiv   pre-print
Motivated by a recent result of Elberfeld, Jakoby and Tantau showing that MSO properties are Logspace computable on graphs of bounded tree-width, we consider the complexity of computing the determinant of the adjacency matrix of a bounded tree-width graph and as our main result prove that it is in Logspace. It is important to notice that the determinant is neither an MSO-property nor counts the number of solutions of an MSO-predicate. This technique yields Logspace algorithms for counting the
more » ... mber of spanning arborescences and directed Euler tours in bounded tree-width digraphs. We demonstrate some linear algebraic applications of the determinant algorithm by describing Logspace procedures for the characteristic polynomial, the powers of a weighted bounded tree-width graph and feasibility of a system of linear equations where the underlying bipartite graph has bounded tree-width. Finally, we complement our upper bounds by proving L-hardness of the problems of computing the determinant, and of powering a bounded tree-width matrix. We also show the GapL-hardness of Iterated Matrix Multiplication where each matrix has bounded tree-width.
arXiv:1412.2470v1 fatcat:thrsngyfq5cydn6uphdmzxute4

Collapsing Exact Arithmetic Hierarchies [chapter]

Nikhil Balaji, Samir Datta
2014 Lecture Notes in Computer Science  
We provide a uniform framework for proving the collapse of the hierarchy, NC 1 (C) for an exact arithmetic class C of polynomial degree. These hierarchies collapses all the way down to the third level of the AC 0hierarchy, AC 0 3 (C). Our main collapsing exhibits are the classes C ∈ {C=NC 1 , C=L, C=SAC 1 , C=P}. NC 1 (C=L) and NC 1 (C=P) are already known to collapse [1, 18, 19] . We reiterate that our contribution is a framework that works for all these hierarchies. Our proof generalizes a
more » ... of from [8] where it is used to prove the collapse of the AC 0 (C=NC 1 ) hierarchy. It is essentially based on a polynomial degree characterization of each of the base classes.
doi:10.1007/978-3-319-04657-0_26 fatcat:m4umv2v5kjda5jjszwjzq67dni

Cyclotomic Identity Testing and Applications [article]

Nikhil Balaji, Sylvain Perifel, Mahsa Shirmohammadi, James Worrell
2021 arXiv   pre-print
We consider the cyclotomic identity testing (CIT) problem: given a polynomial f(x_1,...,x_k), decide whether f(ζ_n^e_1,...,ζ_n^e_k) is zero, where ζ_n = e^2π i/n is a primitive complex n-th root of unity and e_1,...,e_k are integers, represented in binary. When f is given by an algebraic circuit, we give a randomized polynomial-time algorithm for CIT assuming the generalised Riemann hypothesis (GRH), and show that the problem is in coNP unconditionally. When f is given by a circuit of
more » ... ly bounded degree, we give a randomized NC algorithm. In case f is a linear form we show that the problem lies in NC. Towards understanding when CIT can be solved in deterministic polynomial-time, we consider so-called diagonal depth-3 circuits, i.e., polynomials f=∑_i=1^m g_i^d_i, where g_i is a linear form and d_i a positive integer given in unary. We observe that a polynomial-time algorithm for CIT on this class would yield a sub-exponential-time algorithm for polynomial identity testing. However, assuming GRH, we show that if the linear forms g_i are all identical then CIT can be solved in polynomial time. Finally, we use our results to give a new proof that equality of compressed strings, i.e., strings presented using context-free grammars, can be decided in randomized NC.
arXiv:2007.13179v2 fatcat:khek6c2ugfex3lu2sjlkwndp4q

Skew Circuits of Small Width [chapter]

Nikhil Balaji, Andreas Krebs, Nutan Limaye
2015 Lecture Notes in Computer Science  
A celebrated result of Barrington (1985) proved that polynomial size, width-5 branching programs (BP) are equivalent in power to a restricted form of branching programs -polynomial sized width-5 permutation branching programs (PBP), which in turn capture all of NC 1 . On the other hand it is known that width-3 PBPs require exponential size to compute the AND function. No such lower bound is known for width-4 PBPs, however it is widely conjectured that width-4 PBPs will not capture all of NC 1 .
more » ... In this work, we investigate the region inside width-5 branching programs by comparing them with bounded width skew circuits. It is well known that branching programs of bounded width have the same power as skew circuit of bounded width. The naive approach converts a BP of width w to a skew circuit of width w 2 . We improve this bound and show that BP of width w ≥ 5 can be converted to a skew circuit of width 7. This also implies that skew circuits of bounded width are equal in power to skew circuits of width 7. For the other way, we prove that for any w ≥ 2, a skew circuit of width w can be converted into an equivalent branching program of width w. We prove that width-2 skew circuits are not universal while width-3 skew circuits are universal. We show that any polynomial sized CNF or DNF is computable by width 3 skew circuits of polynomial size. We prove that a width-3 skew circuit computing Parity requires exponential size. This gives an exponential separation between the power of width-3 skew circuits and width-4 skew circuits. ACM Subject Classification Dummy classification -please refer to http://www.acm.org/ about/class/ccs98-html
doi:10.1007/978-3-319-21398-9_16 fatcat:7c2ydz3e6zbnjnhkv47lcxvrfa

Cryptanalysis of a Chaotic Image Encryption Algorithm [article]

Nikhil Balaji, Nithin Nagaraj
2008 arXiv   pre-print
Acknowledgements Nikhil Balaji would like to express his sincere gratitude to Department of Electronics and Communication Engineering, National Institute of Technology Karnataka (NITK) where he is an undergraduate  ... 
arXiv:0801.0276v2 fatcat:agwirjxcpjfn3pxywc2evyhcrm

Complexity of Restricted Variants of Skolem and Related Problems

Akshay S., Nikhil Balaji, Nikhil Vyas, Marc Herbstritt
2017 International Symposium on Mathematical Foundations of Computer Science  
Given a linear recurrence sequence (LRS), the Skolem problem, asks whether it ever becomes zero. The decidability of this problem has been open for several decades. Currently decidability is known only for LRS of order upto 4. For arbitrary orders (i.e., number of terms the n th depends on), the only known complexity result is NP-hardness by a result of Blondel and Portier from 2002. In this paper, we give a different proof of this hardness result, which is arguably simpler and pinpoints the
more » ... rce of hardness. To demonstrate this, we identify a subclass of LRS for which the Skolem problem is in fact NP-complete. We show the generic nature of our lower-bound technique by adapting it to show stronger lower bounds of a related problem that encompasses many known decision problems on linear recurrent sequences.
doi:10.4230/lipics.mfcs.2017.78 dblp:conf/mfcs/AkshayBV17 fatcat:vufbz2cgyzd4pfe2xf4yubomhq

Graph properties in node-query setting: effect of breaking symmetry [article]

Nikhil Balaji, Samir Datta, Raghav Kulkarni, Supartha Podder
2015 arXiv   pre-print
The query complexity of graph properties is well-studied when queries are on edges. We investigate the same when queries are on nodes. In this setting a graph G = (V, E) on n vertices and a property P are given. A black-box access to an unknown subset S ⊆ V is provided via queries of the form 'Does i ∈ S?'. We are interested in the minimum number of queries needed in worst case in order to determine whether G[S], the subgraph of G induced on S, satisfies P. Apart from being combinatorially
more » ... this setting allows us to initiate a systematic study of breaking symmetry in the context of query complexity of graph properties. In particular, we focus on hereditary graph properties. The monotone functions in the node-query setting translate precisely to the hereditary graph properties. The famous Evasiveness Conjecture asserts that even with a minimal symmetry assumption on G, namely that of vertex-transitivity, the query complexity for any hereditary graph property in our setting is the worst possible, i.e., n. We show that in the absence of any symmetry on G it can fall as low as O(n^1/(d + 1) ) where d denotes the minimum possible degree of a minimal forbidden sub-graph for P. In particular, every hereditary property benefits at least quadratically. The main question left open is: can it go exponentially low for some hereditary property? We show that the answer is no for any hereditary property with finitely many forbidden subgraphs by exhibiting a bound of Ω(n^1/k) for some constant k depending only on the property. For general ones we rule out the possibility of the query complexity falling down to constant by showing Ω( n/ n) bound. Interestingly, our lower bound proofs rely on the famous Sunflower Lemma due to Erdös and Rado.
arXiv:1510.08267v1 fatcat:m6iegnep2nhbxh5jm63zxfj2q4

Bounded Treewidth and Space-Efficient Linear Algebra [chapter]

Nikhil Balaji, Samir Datta
2015 Lecture Notes in Computer Science  
Motivated by a recent result of Elberfeld, Jakoby and Tantau [EJT10] showing that MSO properties are Logspace computable on graphs of bounded tree-width, we consider the complexity of computing the determinant of the adjacency matrix of a bounded tree-width graph and as our main result prove that it is in Logspace. It is important to notice that the determinant is neither an MSO-property nor counts the number of solutions of an MSO-predicate. This technique yields Logspace algorithms for
more » ... g the number of spanning arborescences and directed Euler tours in bounded tree-width digraphs. We demonstrate some linear algebraic applications of the determinant algorithm by describing Logspace procedures for the characteristic polynomial, the powers of a weighted bounded tree-width graph and feasibility of a system of linear equations where the underlying bipartite graph has bounded tree-width. Finally, we complement our upper bounds by proving L-hardness of the problems of computing the determinant, and of powering a bounded tree-width matrix. We also show the GapL-hardness of Iterated Matrix Multiplication where each matrix has bounded tree-width.
doi:10.1007/978-3-319-17142-5_26 fatcat:hv6yitji25drzd5mpmzer3mwcu

Erasure Coding for Distributed Storage: An Overview [article]

S. B. Balaji, M. Nikhil Krishnan, Myna Vajha, Vinayak Ramkumar, Birenjith Sasidharan, P. Vijay Kumar
2018 arXiv   pre-print
In a distributed storage system, code symbols are dispersed across space in nodes or storage units as opposed to time. In settings such as that of a large data center, an important consideration is the efficient repair of a failed node. Efficient repair calls for erasure codes that in the face of node failure, are efficient in terms of minimizing the amount of repair data transferred over the network, the amount of data accessed at a helper node as well as the number of helper nodes contacted.
more » ... oding theory has evolved to handle these challenges by introducing two new classes of erasure codes, namely regenerating codes and locally recoverable codes as well as by coming up with novel ways to repair the ubiquitous Reed-Solomon code. This survey provides an overview of the efforts in this direction that have taken place over the past decade.
arXiv:1806.04437v1 fatcat:6ivmg6n44jhtjal26btpwjbkpy

Near-Optimal Complexity Bounds for Fragments of the Skolem Problem

S. Akshay, Nikhil Balaji, Aniket Murhekar, Rohith Varma, Nikhil Vyas, Markus Bläser, Christophe Paul
2020 Symposium on Theoretical Aspects of Computer Science  
Balaji, A. Murhekar, R. Varma, and N.  ...  Balaji, A. Murhekar, R. Varma, and N. Vyas 37:9 r and the degrees of each p j (n) ≤ m. where m is a size parameter.  ... 
doi:10.4230/lipics.stacs.2020.37 dblp:conf/stacs/AkshayBMV020 fatcat:k4bhiknnhfbktc7q3i2idjng74

Terms of Lucas sequences having a large smooth divisor

Nikhil Balaji, Florian Luca
2022 Canadian mathematical bulletin  
Balaji and F.  ...  Balaji and F. Luca which gives #(𝑄 2 ∩ (2 Thus, #𝑄 2 ≪ 𝑁 exp − log 𝑁 52 log log 𝑁 𝑦 2 𝑁 .  ... 
doi:10.4153/s0008439522000248 fatcat:cug5xctfzfbltlyj25gw2wd4ei

Counting Euler Tours in Undirected Bounded Treewidth Graphs [article]

Nikhil Balaji, Samir Datta, Venkatesh Ganesan
2015 arXiv   pre-print
We show that counting Euler tours in undirected bounded tree-width graphs is tractable even in parallel - by proving a #SAC^1 upper bound. This is in stark contrast to #P-completeness of the same problem in general graphs. Our main technical contribution is to show how (an instance of) dynamic programming on bounded clique-width graphs can be performed efficiently in parallel. Thus we show that the sequential result of Espelage, Gurski and Wanke for efficiently computing Hamiltonian paths in
more » ... nded clique-width graphs can be adapted in the parallel setting to count the number of Hamiltonian paths which in turn is a tool for counting the number of Euler tours in bounded tree-width graphs. Our technique also yields parallel algorithms for counting longest paths and bipartite perfect matchings in bounded-clique width graphs. While establishing that counting Euler tours in bounded tree-width graphs can be computed by non-uniform monotone arithmetic circuits of polynomial degree (which characterize #SAC^1) is relatively easy, establishing a uniform #SAC^1 bound needs a careful use of polynomial interpolation.
arXiv:1510.04035v2 fatcat:loqv3aauhjds5jmbo5nqcuzwye

Identity Testing for Radical Expressions [article]

Nikhil Balaji, Klara Nosan, Mahsa Shirmohammadi, James Worrell
2022 arXiv   pre-print
Balaji et al.  ... 
arXiv:2202.07961v3 fatcat:ywg7dz7bhjdmvljxquchplhb7q

Tree-width and Logspace: Determinants and Counting Euler Tours [article]

Nikhil Balaji, Samir Datta
2013 arXiv   pre-print
Motivated by the recent result of [EJT10] showing that MSO properties are Logspace computable on graphs of bounded tree-width, we consider the complexity of computing the determinant of the adjacency matrix of a bounded tree-width graph and prove that it is L-complete. It is important to notice that the determinant is neither an MSO-property nor counts the number of solutions of an MSO-predicate. We extend this technique to count the number of spanning arborescences and directed Euler tours in
more » ... ounded tree-width digraphs, and further to counting the number of spanning trees and the number of Euler tours in undirected graphs, all in L. Notice that undirected Euler tours are not known to be MSO-expressible and the corresponding counting problem is in fact #P-hard for general graphs. Counting undirected Euler tours in bounded tree-width graphs was not known to be polynomial time computable till very recently Chebolu et al [CCM13] gave a polynomial time algorithm for this problem (concurrently and independently of this work). Finally, we also show some linear algebraic extensions of the determinant algorithm to show how to compute the charcteristic polynomial and trace of the powers of a bounded tree-width graph in L.
arXiv:1312.7468v2 fatcat:cfyu5jvox5hnzjyinydlsxethi

Gastrointestinal Hemorrhage in Patient with Granulomatosis with Polyangitis

Nikhil Madan, Vipul Patel, Balaji Yegneswaran
2021 Case Reports in Critical Care  
Granulomatosis with polyangitis (GPA) is characterized by a necrotizing granulomatous vasculitis of small arteries and veins. It most commonly affects the upper and lower respiratory tract and kidneys. However, other organs including the gastrointestinal tract can be affected. Gastrointestinal manifestations of GPA are rare and can include ischemia, bowel infarction, and perforation. Hemorrhage is an extremely rare presentation of GPA. We present a case of a woman with GPA and pulmonary renal
more » ... ndrome on treatment who presents with severe gastrointestinal hemorrhage.
doi:10.1155/2021/9921361 pmid:34737897 pmcid:PMC8563110 fatcat:ttvznxv6o5gcpnqsjnoz43tktm
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